Graphing Rational Functions by Hand
*A, B, and C objectives below are objectives that most students have "had"
and understand a some level (however, often understanding is weak). By going
through this activity of learning how to graph rational functions by hand, the
understanding of A, B, and C objectives are strengthened.
The D objectives are objectives that the student will learn as a result
of doing this activity (you probably do not know these objectives yet). These
are useful objectives about functions, rational functions, and graphing.
- *Basic and Intermediate Number Sense Concepts
- *Basic Function Sense
- *Procedural Skills
- Advanced Function Sense
- *Basic and Intermediate Number Sense Concepts ~ The student should
- Zero times a number is zero.
- Zero plus a number is the number.
- A fraction is zero when the numerator (top) is zero.
- A fraction is undefined (not defined) when the denominator (bottom)
- pos./pos. = pos.; neg./neg. = pos.; neg./pos. = neg.; pos./neg. = neg.
(where "pos." stands for a positive number, and "neg."
stands for a negative number).
- When you divide a number by a small number (less than 1) the answer
is bigger than the original number.
- When you divide a number by a tiny number (close to zero) the answer
is large (the smaller the divisor, the larger the answer).
- *Basic Function Sense ~ The student should understand that:
- When plugging zero into a polynomial, you get the constant term.
- To find the x-intercept, put zero in for y and solve.
- To find the y-intercept, put zero in for x and solve.
- If a polynomial is factored, and (x-a) is a factor, "a"
being a number, then if you plug in x=a, then the result is zero.
- *Procedural Skills ~ The student should be able to:
- Plug zero into a polynomial that is in expanded form (use objective B.1).
- Plug zero into a polynomial that is in factored form.
- Plug zero into a rational function.
- Advanced Function Sense ~ The student should understand
- When the denominator is zero (for a rational function) it usually means
there is a vertical asymptote at the value that causes the denominator to
- If (x-a) is a factor of the denominator, "a" being
a number, if one puts in x-values very close to a, then the
denominator is very small, usually causing the function to have a value
which is large positive or large negative.
On either side of a vertical asymptote the graph commonly goes up "to
positive infinity" or "down to negative infinity" (which
means that the function values grow large without bound, or become largely
negative without bound).
For a rational function, as the input values grow large ("approach
positive or negative infinity"), the function values (value of the
fraction) might do one of three things:
- Approach 0 (this happens when the denominator [bottom] dominates the
- Keep growing (positively or negatively - this happens when the numerator
[top] dominates the fraction).
- Approach a specific number (this happens when neither the numerator
or the denominator dominates the fraction).
- For a polynomial, the term with the highest degree will dominate the polynomial.
- For a rational function, the term with the highest degree will dominate
the rational function, if there is one.
Reasons Why This is a Worthwhile Activity
- The activity brings together number sense, function sense, and graph sense.
The primary strength of the activity is the connections is builds.
- While the process my appear procedural, it is largely a conceptual process
of finding relationship--rather like putting a puzzle together.
- Helps one realize that all functions are not nice lines or curves without
breaks. Function graphs can have breaks.
- Helps one consider "global" features of graphs, not just pointwise
- By using basic skills as part of a bigger problem, one better learn the
- Students often have difficulty with asymptotes (not just the spelling).
By graphing rational functions by hand, one learns what the asymptotes mean.
- Helps one realize that sometimes one can learn more by graphing by hand
than by using technology.
What Types of Rational Functions are We Graphing Here?
To keep the amount of arithmetic low, and allow for the graphs to be constructed
conceptually, the following is true of the rational functions in this activity.
- Numerators and denominators are polynomials, frequently linear and with
- Usually numerators and denominators of degree two or higher are factorable
(or already factored).
- Generally, the pathological same-factor-in-both-the-numerator-and-denominator
is not here (but that can be added later).
- Generally, I am in favor of using a graphics calculator (or other technology)
for graphing functions. However, in this case, producing by-hand graphs is
highly desirable. They can be produced quite easily (with very little computational
arithmetic) and the process of producing the graphs helps one understand many
concepts related to functions.
- Graphics calculators (graphs and tables) may be useful in the
learning process. However, the intention is that they are only tools to
help the student learn the concepts. Once the concepts are understood, the
graphics calculator is no longer needed and should no longer be used.
- Generally, I am in favor of lessons that have real-world applications. This
lesson is a counterexample to that principle as well. This activity requires
abstract thinking, without a lot of computational arithmetic.
- Ironically, an activity that appears to be procedural in nature turns out
to be quite conceptual.
© James Olsen, Western
Illinois University Mathematics Department
March 19, 2008